
\prob{004B}{Fibonacci数列}

求Fibonacci数列

\[ F_n = \begin{cases}
  1 & n = 0, 1 \\
  F_{n - 1} + F_{n - 2} & n > 1 \\
\end{cases} \]

的通项公式。
\problabels{yellow/代数, green/代数求值问题}

\ans{
  \[ F_n = \frac{\sqrt5}5\left(\left(\frac{1 + \sqrt5}2\right)^n - \left(\frac{1 - \sqrt5}2\right)^n\right) \]
}
